\chapter{Verifying a Field: The Full Campaign}
\label{ch:field}

\section{The summit statement}

Every thread so far --- specs as types, tactics, automation, $\Fp$,
primality, extraction, denotation --- was preparation for one theorem. In
the companion repositories it is called the \emph{field implementation
certificate}, and (lightly paraphrased) it says:

\begin{lstlisting}[language=Lean]
theorem fieldImplementation :
    -- p is prime, so ZMod p is genuinely a field
    Nat.Prime p
    -- and for all bounded limb arrays, every operation's
    -- commuting square closes:
  ∧ (∀ a b, Bnd a → Bnd b → AddSquare a b)
  ∧ (∀ a b, Bnd a → Bnd b → SubSquare a b)
  ∧ (∀ a b, Bnd a → Bnd b → MulSquare a b)
  ∧ (∀ a,   Bnd a →         SquareSquare a)
  ∧ (∀ a,   Bnd a →         InvertSquare a)   -- Fermat chain: a^(p-2)
  ∧ ...                                       -- reduce, negate, encode
\end{lstlisting}

One theorem, kernel-checked, quantified over \emph{every} input the
representation admits: the extracted dalek field arithmetic implements
$\Fp$. This chapter is the story of the campaign that proves it --- told
honestly, including the two places where the terrain fought back, because
the failures teach more than the victories.

\section{Order of battle}

You do not prove such a conjunction by heroism; you prove it by sequencing.
The campaign order in the real projects, and the reason for each position:

\begin{enumerate}[leftmargin=1.6em]
\item \textbf{Bounds lemmas first} --- pure \lean{omega} facts about limb
  sizes, no denotation at all. Cheap, and everything depends on them.
\item \textbf{Primality} (Chapter~\ref{ch:prime}) --- independent of the
  code entirely; it dignifies \lean{ZMod p} into a field.
\item \textbf{add, sub, negate} --- linear operations; the commuting squares
  close with \lean{omega} plus the denotation unfolds. Confidence builders.
\item \textbf{reduce} --- the $\times 19$ fold in isolation. Proving it
  alone, before mul uses it, halves the hardest proof's size.
\item \textbf{mul, square} --- the boss fight of Chapter~\ref{ch:denotation}:
  column sums, carries, folds. Squaring is mul with algebraic shortcuts ---
  a separate code path in dalek, hence a separate theorem. No shortcuts
  in the proof: the \emph{code's} shortcut is precisely what needs checking.
\item \textbf{invert} --- the 254-squaring Fermat chain, verified as a
  \lean{calc} of exponent bookkeeping on top of \code{mul}/\code{square}
  specs, ending at $a^{p-2}$; Fermat's little theorem (Mathlib's) closes
  the square.
\end{enumerate}

Notice the shape: \emph{each layer consumes only the specs of the layer
below}, never reaching into implementations. By step 6 you are doing exponent
arithmetic, blissfully ignorant of carries. That is the compositionality that
Chapter~\ref{ch:denotation}'s two-clause specs (bounds + value) were designed
to buy.

\section{Dispatches from the terrain}

\textbf{The wall that was really there.} Partway up, one proof style hit a
genuine limit of the tool: correctness certificates for \code{mul}-scale
goals, when handed to a general decision procedure in one monolithic call,
generate internal certificates with coefficients on the order of $2^{256}$
--- and checking them can exhaust the proof checker's memory. One such call,
during the development of the Pasta field proofs, consumed twelve gigabytes
and crashed the machine (Chapter~\ref{ch:automation} told you this story
from the tactic side). The cure was never cleverness --- it was
\emph{decomposition}: isolate each carry step as its own small lemma with a
tiny context, prove the value identity with \lean{linear_combination}
(a tactic that checks a \emph{stated} linear certificate rather than
searching for one), and let the big theorem be an assembly of small
checked parts. The same discipline, plus hard memory caps on the checker
process, became house infrastructure.

\begin{aha}
There is a deep symmetry in that fix worth savoring: the \emph{proof} was
restructured exactly the way the \emph{code} was --- into small steps with
controlled intermediate size. Delayed carries in the implementation; small
lemmas in the verification. Bounded limbs; bounded contexts. Good proofs and
good fast code turn out to obey the same engineering aesthetics. This is not
a coincidence; both are fighting combinatorial growth with structure.
\end{aha}

\textbf{Four forks, one method, real divergence.} The companion projects
verify not just upstream \code{curve25519-dalek} but three production forks
(Solana's, RISC~Zero's, Betrusted's) --- each against \emph{its own}
extraction. Worth it? The audit found the forks implement the same
constant-time conditional selection three different ways (a \code{subtle}
crate trait, a volatile-read \code{black_box}, a hand-rolled arithmetic
mask), and one fork reorders instructions in point doubling. All correct ---
\emph{provably}, now --- but the divergence is exactly the kind of thing
that silently breaks when someone ``harmonizes'' code during a rebase.
Per-fork verification is not pedantry; it is how you notice that ``the same
library'' isn't.

\begin{pitfall}
A verified fork is verified \emph{at a commit}. Change one line of
arithmetic and the certificate is stale --- that is a feature (the proof
\emph{should} break when the code changes), but it means verification is a
\emph{process wired into maintenance}, not a trophy. The companion repos
ship \code{check.sh} scripts that re-extract and re-verify from scratch;
treat those as the project's pulse, not as CI decoration.
\end{pitfall}

\section{Reading a certificate like a professional}

Suppose a stranger hands you a repository claiming ``formally verified
field arithmetic.'' Chapter~\ref{ch:honesty} gives you the full audit
protocol, but the field-layer questions you can already ask are these:

\begin{itemize}[leftmargin=1.4em]
\item \textbf{What is denoted?} Find the denotation function. Does it map
  the \emph{extracted} representation (good) or a hand-written lookalike
  (Chapter~\ref{ch:rust}'s hole)?
\item \textbf{Is the square complete?} Value equation \emph{and} bounds
  propagation \emph{and} the \lean{.ok} clause --- a spec that assumes
  success proves nothing about overflow.
\item \textbf{Is the quantifier honest?} \lean{∀ a b} with bounds
  hypotheses, or a handful of \lean{example}s on constants dressed up as
  coverage?
\item \textbf{Does anything say \lean{sorry}?} One \lean{sorry} anywhere in
  the dependency chain and the certificate is decorative. The checker will
  tell you; ask it.
\end{itemize}

\begin{tryit}
Do the audit for real: open \code{dalek-ed25519-verified}, find the
denotation, pick the \code{sub} spec, and check the three clauses against
the list above. Then run the repo's \code{check.sh} and watch the whole
pyramid rebuild. Total time: one coffee. Skill acquired: permanent.
\end{tryit}

\section*{Exercises}

\exercise{Field subtraction in dalek computes $a - b$ as
$a + (16p - b)$ limb-wise (adding a multiple of $p$ keeps limbs positive ---
recall \lean{Nat} truncation!). State the commuting square for \code{sub}
including the exact multiple-of-$p$ fact you would need as a lemma, and
explain why $16p$ rather than $p$.}

\exercise{The Fermat inversion chain computes $a^{p-2}$ via 254 squarings
and 11 multiplications. Verify the exponent bookkeeping for the first three
steps of dalek's actual chain: $a^2$, $a^{9} = (a^2)^{2\cdot 2} \cdot a$,
$a^{11} = a^9 \cdot a^2$. (The full chain is exercise-by-induction: each
step's exponent is a sum of previous ones --- addition-chain arithmetic.)}

\exercise{(Discussion) Step 5 refuses to trust the squaring shortcut and
verifies \code{square} separately from \code{mul}. A colleague argues
``square is just \code{mul a a}, reuse the theorem.'' What, concretely,
would that argument miss about the code under verification?}

\begin{checkpoint}
You should now be able to: state the field implementation certificate and
every quantifier in it; recite the campaign order and justify why bounds
and \code{reduce} come early; tell the memory-wall story and its
decomposition moral; and audit a stranger's field-layer claim with four
pointed questions.
\end{checkpoint}